M^ω considered as a programming language

Abstract

The paper studies a simply typed term system M^ω providing a primitive recursive concept of parallelism in the sense of Plotkin. The system aims at defining and computing partial continuous functionals. Some connections between denotational and operational semantics → for M^ω are investigated. It is shown that → is correct with respect to the denotational semantics. Conversely, → is complete in the sense that if a program denotes some number k, then it is reducible to the numeral n_k.

Restricting to the primitive recursive kernel $\text{℘R}{}^{\mathrm{\omega }}$ of M^ω, it is shown that → is strongly normalising with uniquely determined normal forms. The twist is the design of fixed point style conversion rules for constants μ_$\text{l}$ accounting for parallelly bounded parallel search such that correctness and strong normalisation hold. Thereupon, minor alternations to → bring about that every reduction sequence for a program of $\text{℘R}{}^{\mathrm{\omega }}$ terminates either in a numeral n_k if the program denotes k, or in the term (⊂-1)0 if the program denotes the “undefined” object. Thus, $\text{℘R}{}^{\mathrm{\omega }}$ can be considered a primitive recursive version of Plotkin's $\text{L}$_PA+∃·.